Symmetry Breaking in Graphs Motion Conjectures Wilfried Imrich Montanuniversit¨ at Leoben, Austria

Joint Mathematics Meetings, Baltimore, Maryland My Favorite Graph Theory Conjectures, III January 17, 2014

ΠΑΝΤΑ ΡΕΙ Everything Flows Maybe the First Recorded Motion Conjecture Attributed to Heraklit

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Conjectures about Motion and Symmetry Breaking The conjectures concern Symmetries of Countable or Uncountable Infinite Graphs and Groups and how automorphisms or endomorphisms are broken by deterministic or random colorings.

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An Example for Symmetry Breaking in a Finite Graph Consider the given 3-coloring of the vertices of the 3-cube Q3:

Only the identity automorphism of Q3 preserves the colors. We say Q3 is 3-distinguishable. 3

Definition Let A be a group acting on a set X. Then - a coloring C of the elements of X is distinguishing if |AC | = 1. - If C uses d colors, then C is d-distinguishing. - The smallest such d is the distinguishing number D(A, X) of (A, X). (Albertson and Collins, 1996) - The distinguishing number D(G) of a graph G is D(Aut(G), V (G)). D(Kn) = n. The distinguishing number of asymmetric graphs is 1. Conjecture Almost all graphs with non-trivial automorphism group are 2-distinguishable. 4

More Examples and Motion

C6 and Pℵ0 are 2-distinguishable, their motion is 4, resp. ℵ0. 5

D(G) and the Motion of a Graph The motion m(α) of α ∈ A, where A acts on a set X, is the number of elements it moves. The motion of a graph G is m(G) = minα∈Aut(G)\{id} m(α). m(Kn) = 2, m(C6) = 4, m(Pℵ0 ) = ℵ0. Russell and Sundaram 1998: Lemma (Motion Lemma) If is d-distinguishable.

m(G) d 2

≥ |Aut(G)| , then the graph G

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1. The Infinite Motion Conjecture (Tucker 2011) Let G be an infinite, locally finite, connected graph with infinite motion. Then D(G) = 2. The conjecture holds, for example, for graphs with countable automorphism group and infinite homogeneous trees; the proofs are easy. The strongest result is: Theorem (Lehner) The Infinite Motion Conjecture is true for graphs of growth (

O

√ ) n (1−ϵ) 2 2

The proof extends methods of Cuno, Imrich and Lehner. It partitions automorphisms into classes with different motion. Johannes Cuno and Florian Lehner are PhD students at the Doctoral Program Discrete Mathematics at the TU & KFU Graz and MU Leoben. 7

2. The Probabilistic Infinite Motion Conjecture (Lehner) Let G be an infinite, locally finite, connected graph with infinite motion. Then a random coloring of G is almost surely distinguishing. It holds in all those cases, where the IMC has been shown to hold. Notice that random colorings of D(Tℵ0 ) are almost surely not distinguishing, but D(Tℵ0 ) = 2. Lehner and Moeller have examples of connected, infinite graphs of countable degree and infinite motion with infinite distinguishing number. 8

The Permutation Topology Every permutation group A acting on a countable set X carries a natural topology, the permutation topology. It is defined by choosing as open sets the cosets of the pointwise stabilizers of the finite subsets of X. Notice that Aut(G) is closed in Sym(V (G)) if G is a connected, locally finite infinite graph∗. Also, the orbits of every stabilizer Aut(G)v , where v ∈ V (G), are finite. Such groups are called subdegree finite. ∗A

permutation group A on a set is closed if and only if A is the full automorphism group of some structure on that set, say a graph. 9

3. The Infinite Motion Conjecture for Permutation Groups (Imrich, Smith, Tucker, Watkins) If A is a closed, subdegree-finite permutation group with infinite motion on a countably infinite set X, then D(A, X) = 2. For countable A one needs neither closure nor subdegree-finiteness. Notice that topological methods are essential for the proof that the Probabilistic IMC holds for graphs of intermediate growth. Lehner begins his proof by showing that every subdegree-finite, closed permutation group A on a set X is totally disconnected, locally compact, and Polish∗. ∗ homeomorphic

to a complete metric space with a countable dense subset. 10

Theorem The distinguishing number of the homogeneous tree Td of degree d > 2 is 2. Proof for d = 4:

The result D(T3) = 2 is already due to Babai 1977. Essentially the same proof also holds for d = ℵ0. 11

Recall, to prove the conjectures we have to show that D(A, X) = 2. This is equivalent to showing that ∃ a 2-coloring C with Ac = id. Theorem (Tucker) Let A ≤ Sym(X), where X is countable. m(A) = ℵ0, then A has a dense 2-distinguishable subgroup B.

If

Hence ∃ a 2-coloring C such that AC is completely disconnected. Theorem (Lehner) Let G be a locally finite infinite graph with infinite motion and let C be a random coloring of G. Then (Aut G)C is almost surely a nowhere dense, closed subgroup with Haar measure 0. Theorem (Lehner) Let G be a locally finite infinite graph with infinite motion. Then there exists a coloring C of G such that (Aut G)C has measure 0 in Aut G and is nowhere dense and compact. 12

4. The Motion Conjecture for Uncountable Graphs (Cuno, Imrich, Lehner) Let G be a connected graph of uncountable cardinality with infinite motion m(G) and suppose that 2m(G) ≥ |Aut(G)|. Then D(G) = 2. It is an immediate generalization of the Motion Lemma: 2m(G)/2 ≥ |Aut(G)|

implies

D(G) = 2 .

Theorem Let G be a connected graph with uncountable motion. Then |Aut(G)| ≤ m(G) implies D(G) = 2. Corollary Let G be a connected graph with uncountable motion. If the general continuum hypothesis holds, and if |Aut(G)| < 2m(G), then D(G) = 2. 13

The Motion Conjecture for Uncountable Graphs holds in the following cases D(Qn) = 2, where Qn is the hypercube of dimension n. Transfinite induction. D(Kn2Kn) = 2. Transfinite induction. It does not hold for connected countable graphs that are not locally finite. For such graphs we have: D(R) = 2, where R is the countable random graph. Tricky proof. D(Kℵ0 2Kℵ0 ) = 2, where 2 denotes the Cartesian product. Easy. D(Tℵ0 ) = 2, we already discussed this case. 14

The following result are outside the scope of the Motion Conjecture for Uncountable Graphs D(Kn2K2n ) = 2. Transfinite induction. n

This graph has motion n and |Aut(Kn2K2n )| = 22 > 2n. This result is not implied by the Motion Conjecture for Uncountable Graphs. D(Kn2Km) > n if m > 2n. Here too m(Kn2Km) = n , but |Aut(Kn2Km)| = 2m > 2n. This result is in accordance with the Motion Conjecture for Uncountable Graphs. 15

Endomorphisms∗ Definition Let G be a graph. A coloring C of V (G) with d colors is endomorphism d-distinguishing, if | End(G)c| = 1.

De(Q3) = 3 Clearly D(G) ≤ De(G). If End(G) = Aut(G), then De(G) = D(G). ∗ Imrich,

Kalinowski, Lehner and Pilsniak, Endomorphism Breaking in Graphs, E-JC, accepted for publication. 16

Notice that De(G) can be equal to D(G) even when Aut(G) ⊂ End(G). For example, De(C2k ) = 2 for k ≥ 3.

v5 v2k−1

v4

v2k

v3 v1

v2

If Aut(G) is trivial, but 2 ≤ | End(G)|, then D(G) < De(G). Examples are asymmetric, nontrivial finite trees T . The endomorphism distinguishing number of finite paths is two. 17

The endomorphism motion of a graph G is me(G) =

min

m(ϕ)

ϕ∈End(G)\{id}

For example, me(C4) = 1, me(C5) = 4, me(C100) = 49, me(K100) = 2. Lemma (Endomorphism Motion Lemma) If me (G) d 2

≥ | End(G)| ,

then G is endomorphism d-distinguishable.

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5. The Infinite Endomorphism Motion Conjecture Let G be a connected, locally finite infinite graph with infinite endomorphism motion. Then De(G) = 2. A tree has infinite endomorphism motion if and only if it has no pendant vertices. Theorem The endomorphism distinguishing number of countable trees T without pendant vertices is 2. This includes rays (one-sided infinite paths) and two-sided infinite paths. Notice that these graphs have uncountable endomorphism monoids. 19

Theorem Let G be a graph with infinite me(G) and countable End(G). Then De(G) = 2. It is not hard to see that finitely generated groups have countable endomorphisms monoids and infinite endomorphism motion. Theorem If Γ is a finitely generated group. Then there is a twocoloring of Γ, such that the identity endomorphism of Γ is the only endomorphism that preserves this coloring. In other words, De(Γ) = De(End(Γ), Γ) = 2. This generalizes a result of Tucker for automorphisms of finitely generated groups. 20

6. The Endomorphism Motion Conjecture for Uncountable Graphs Let G be a connected, infinite graph with endomorphism motion me(G). If 2me(G) ≥ | End(G)|, then De(G) = 2. Theorem Let G be a connected graph with uncountable endomorphism motion. Then |Aut(G)| ≤ me(G) implies De(G) = 2. Corollary Let G be a connected graph with uncountable endomorphism motion. If the general continuum hypothesis holds, and if | End(G)| < 2me(G), then De(G) = 2.

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Thanks Special thanks to Sandi Klavˇ zar, who introduced me to the subject, and to Thomas Tucker, who urged me to have a look at his Infinite Motion Conjecture. I am grateful to Johannes Cuno and Florian Lehner for their enthusiasm and mathematical contributions, to Simon Mark Smith, Tom Tucker and Mark Watkins for the interaction across the Atlantic, to Vladimir Trofimov for joint work on infinite graphs, and to Rafal Kalinowski and Monika Pilsniak for cooperation on the endomorphim distinguishing number. 22